Image sampling and quantization

1. Image Sampling and Quantization
Subject: Image Procesing & Computer Vision
Dr. Varun Kumar
Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 3 1 / 16

2. Outlines
1 2D Sampling
2 Spectrum of an Image
3 Quantization
4 Optimum Mean Square or Lloyd-Max Quantizer
5 Design an Optimum Quantizer by Signal PDF
6 References
Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 3 2 / 16

3. Previous Discussion
1D Sampling
x(n) = xs(t) = x(t)comb(t, ∆Ts) =
∞
n=−∞
x(t)δ(t − n∆Ts) (1)
By the properties of convolution:
x1(t)x2(t) ⇐⇒ X1(Ω) ⊗ X2(Ω)
x1(t) ⊗ x2(t) ⇐⇒ X1(Ω)X2(Ω)
Spectrum of sampled signal:
Xs(Ω) = X(Ω) ⊗ F(comb(t, ∆Ts)) (2)
Note: For successful recovery of original signal from the sampled signal,
when Ωs ≥ 2Ω0 (Nyquist criteria)
when Ωs < 2Ω0 (Aliasing)
Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 3 3 / 16

4. 2D Sampling
2D Sampling
In 1D sampling time domain signal is sampled in cycle/unit time.
In 1D sampling, signal X(Ω) is band-limited ⇒
X(Ω) = 0 ∀ |Ω| > Ω0
In 2D sampling image is sampled in cycle/unit length in x-y direction.
In 2D sampling, image signal G(Ωx , Ωy ) is band-limited ⇒
G(Ωx , Ωy ) = 0 ∀ |Ωx | > Ωx0 and |Ωy | > Ωy0
Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 3 4 / 16

5. Spectrum of Bandlimited 1D and 2D Signal
2D Sampled Signal
gs(x, y) = g(x, y)comb(x, y; ∆x, ∆y)
=
∞
m=−∞
∞
n=−∞
g(x, y)δ(x − m∆x)δ(y − n∆y)
(3)
Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 3 5 / 16

6. Building Block of 2D Digital Image
Spectrum of 2D Sampled Signal
Gs(Ωx , Ωy ) = G(Ωx , Ωy ) ⊗ COMB(Ωx , Ωy ) (4)
Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 3 6 / 16

7. Continued–
where
COMB(Ωx , Ωy ) = F(comb(x, y; ∆x, ∆y))
=ΩxsΩys
∞
m=−∞
∞
n=−∞
δ(Ωx − mΩxs, Ωy − nΩys)
=ΩxsΩyscomb Ωx , Ωy ;
1
∆x
,
1
∆y
(5)
Here, Ωxs = 1
∆x = Sampling frequency along x-direction and
Ωys = 1
∆y = Sampling frequency along y-direction
Note: For proper reconstruction of image from the sampled data is only
possible, when
Ωxs > 2Ωx0 (6)
Ωys > 2Ωy0 (7)
Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 3 7 / 16

8. Spectrum of 2D and Sampled 2D signal
Recovery of an Image from Sampled 2D Signal
Let a low pass filter is designed in such a way that
H(Ωx , Ωy ) =
1
ΩxsΩys
∀ − Ωx0 < Ωx < Ωx0 and − Ωy0 < Ωx < Ωy0
= 0, otherwise
(8)
Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 3 8 / 16

9. Continued–
Hence, the original 2D signal can be recovered from the sampled 2D
signal, when it passes through the low pass filter. Mathematically,
G(Ωx , Ωy ) = Gs(Ωx , Ωy )H(Ωx , Ωy ) (9)
Image Reconstruction Result
1. Higher the value of sampling frequency Ωxs and Ωys greater be the
resolution of image.
2. Due to lower sampling frequency there is chance of poor resolution of
image or blur image may be observed.
3. Image quality is also measured by dots per inch (dpi). Higher the dpi
greater be the sampling frequency.
Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 3 9 / 16

10. Image Quantization
Quantization is a mapping of continuous variable into C to discrete
variable D.
D ∈ {d1, d2, ......, dL}
Mapping is a stair case function.
Quantization rule:
Let we define a set of transition L + 1 transition level, such that
Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 3 10 / 16

11. Continued–
C ∈ {t1, t2, ......tL+1} (10)
t1 → Minimum level
tL+1 → Maximum level.
D = dk if tk < C < tk+1
Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 3 11 / 16

12. Quantization Error:
Lloyd-Max Quantizer:
1 This quantizer minimize the mean square error for a given number of
quantization level.
2 Let C be a real scaler random variable with continuous probability
density pC (c).
3 It is desired to find the decision or transition level tk and the
reconstruction level dk for a L-level quantizer.
Mean Square Error
ζ = E((C − D)2
) =
tL+1
t1
(C − D)2
pC (c)dC
=
tL+1
t1
(C − dk)2
pC (c)dC
(11)
Note: Our aim is to minimize this error.
Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 3 12 / 16

13. Quantizer Design:
Differentiating (11) wrt tk and dk and equating to 0.
∂ζ
∂tk
= (tk − dk−1)2
pC (tk) − (tk − dk)2
pC (tk) = 0 (12)
∂ζ
∂dk
= 2
tL+1
t1
(C − dk)pC (c)dC = 0 ∀ 1 < k < L (13)
Since, tk < tk−1
tk =
dk + dk−1
2
(14)
rk =
tk+1
tk
CpC (c)dC
tk+1
tk
pC (c)dC
(15)
Note: When the number of quantization level is large, an approximate
solution can be obtained by modeling the pC (c) as piece wise constant.
Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 3 13 / 16

14. Continued–
Using approximate solution for decision level is obtained as
tk+1 = A
A
zk +t1
t1
pC (c)
−1
3 dC
tL+1
t1
pC (c)
−1
3 dC
(16)
where A = tL+1 − t1 and zk = k
L A ∀ k = 1, 2, ....L
t1 and tL+1 both are finite that determines the dynamic range of
quantizer A.
Quantizer mean square distortion is obtained as
ζ =
1
12L2
tL+1
t1
pC (c)
−1
3 dC
3
(17)
Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 3 14 / 16

15. Continued–
Commonly used densities for quantization of image related data are
1 Gaussian
pC (c) =
1
√
2πσ2
exp −
(c − c0)2
2σ2
(18)
2 Laplacian
pC (c) =
1
σ2
exp −
(c − c0)
σ2
(19)
where c0 and σ2 shows the mean and variance.
3 Uniform
pC (c) =
1
tL+1 − t1
∀ t1 < c < tL+1
= 0 otherwise
(20)
Here, uniform quantizer is not very popular.
Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 3 15 / 16

16. References
M. Sonka, V. Hlavac, and R. Boyle, Image processing, analysis, and machine vision.
Cengage Learning, 2014.
D. A. Forsyth and J. Ponce, “A modern approach,” Computer vision: a modern
approach, vol. 17, pp. 21–48, 2003.
L. Shapiro and G. Stockman, “Computer vision prentice hall,” Inc., New Jersey,
2001.
R. C. Gonzalez, R. E. Woods, and S. L. Eddins, Digital image processing using
MATLAB. Pearson Education India, 2004.
Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 3 16 / 16